Putnam TNG - Functional Equations & Iteration
1: Let f: R2 → R be a function such that f(x, y) + f(y, z) + f(z, x) = 0 for all real numbers x, y, and z. Prove that there exists a function g: R → R such that f(x, y) = g(x) - g(y) for all real numbers x and y.
2: Find all functions f: R+ → R such that for all x, y ∈ R+,
f(x + y) = f(x2 + y2).
Here, R+ is the set of positive real numbers.
3: Determine all real functions f that satisfy
f(x2 - y2) = xf(x) - yf(y)
for all pairs of real numbers x and y.
4: Let f(x) be a continuous function such that f(2x2 - 1) = 2xf(x) for all x. Show that f(x) = 0 for -1 ≤ x ≤ 1.
5: Let S denote the set of rational numbers different from {-1, 0, 1}. Define f: S → S by f(x) = x - 1/x. Prove or disprove that
n=1∩∞f(n)(S) = ∅,
where f(n) denotes f composed with itself n times.
6: Let f(x) be a function defined on x > 0 such that f(x) > 0. Find all functions that satisfy: f(x)f(yf(x)) = f(x + y), for all x, y > 0.
7: An (ordered) triple (x1, x2, x3) of positive irrational numbers with x1 + x2 + x3 = 1 is called balanced if each xi < 1/2. If a triple is not balanced, say xj > 1/2, one performs the balancing act
B(x1, x2, x3) = (x'1, x'2, x'3)
where x'i = 2xi if i ≠ j and x'j = 2xj - 1. If the new triple is not balanced, one performs the balancing act on it again. Does continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act?
8: Let F: [0, 1] → [0, 1] be a continuous function such that f(f(f(x))) = x for all x ∈ [0, 1]. Prove that f(x) = x for all x ∈ [0, 1].